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Math  /  Geometry

Question1. Which of the following points is equidistant from A(−2,7)A(-2,7) and B(−6,−5)B(-6,-5) ? A. (0,−4)(0,-4) B. (7,−10)(7,-10) C. (−1,0)(-1,0) D. (4,−5)(4,-5)

Studdy Solution

STEP 1

1. The points A(−2,7) A(-2, 7) and B(−6,−5) B(-6, -5) are given.
2. We need to find which point from the options is equidistant from both A A and B B .

STEP 2

1. Recall the distance formula between two points.
2. Calculate the distance from each option to point A A .
3. Calculate the distance from each option to point B B .
4. Compare the distances to determine which point is equidistant from A A and B B .

STEP 3

Recall the distance formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2):
d=(x2−x1)2+(y2−y1)2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

STEP 4

Calculate the distance from each option to point A(−2,7) A(-2, 7) .
For point (0,−4) (0, -4) :
dA=(0−(−2))2+(−4−7)2=22+(−11)2=4+121=125 d_A = \sqrt{(0 - (-2))^2 + (-4 - 7)^2} = \sqrt{2^2 + (-11)^2} = \sqrt{4 + 121} = \sqrt{125}
For point (7,−10) (7, -10) :
dA=(7−(−2))2+(−10−7)2=92+(−17)2=81+289=370 d_A = \sqrt{(7 - (-2))^2 + (-10 - 7)^2} = \sqrt{9^2 + (-17)^2} = \sqrt{81 + 289} = \sqrt{370}
For point (−1,0) (-1, 0) :
dA=(−1−(−2))2+(0−7)2=12+(−7)2=1+49=50 d_A = \sqrt{(-1 - (-2))^2 + (0 - 7)^2} = \sqrt{1^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50}
For point (4,−5) (4, -5) :
dA=(4−(−2))2+(−5−7)2=62+(−12)2=36+144=180 d_A = \sqrt{(4 - (-2))^2 + (-5 - 7)^2} = \sqrt{6^2 + (-12)^2} = \sqrt{36 + 144} = \sqrt{180}

STEP 5

Calculate the distance from each option to point B(−6,−5) B(-6, -5) .
For point (0,−4) (0, -4) :
dB=(0−(−6))2+(−4−(−5))2=62+12=36+1=37 d_B = \sqrt{(0 - (-6))^2 + (-4 - (-5))^2} = \sqrt{6^2 + 1^2} = \sqrt{36 + 1} = \sqrt{37}
For point (7,−10) (7, -10) :
dB=(7−(−6))2+(−10−(−5))2=132+(−5)2=169+25=194 d_B = \sqrt{(7 - (-6))^2 + (-10 - (-5))^2} = \sqrt{13^2 + (-5)^2} = \sqrt{169 + 25} = \sqrt{194}
For point (−1,0) (-1, 0) :
dB=(−1−(−6))2+(0−(−5))2=52+52=25+25=50 d_B = \sqrt{(-1 - (-6))^2 + (0 - (-5))^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50}
For point (4,−5) (4, -5) :
dB=(4−(−6))2+(−5−(−5))2=102+02=100 d_B = \sqrt{(4 - (-6))^2 + (-5 - (-5))^2} = \sqrt{10^2 + 0^2} = \sqrt{100}

STEP 6

Compare the distances to determine which point is equidistant from A A and B B .
From the calculations: - Point (−1,0) (-1, 0) has distances 50 \sqrt{50} from both A A and B B .
Thus, the point that is equidistant from A(−2,7) A(-2, 7) and B(−6,−5) B(-6, -5) is:
(−1,0) \boxed{(-1, 0)}

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